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        <blockquote>
<p>最长公共子序列是一个基本的dp问题</p>
</blockquote>
<h3 id="1-状态转移方程"><a href="#1-状态转移方程" class="headerlink" title="1.状态转移方程"></a>1.状态转移方程</h3><p><img src="/blogimg/suanfa/1.png" alt=""></p>
<blockquote>
<p>java代码</p>
</blockquote>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> java.util.Scanner;</span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Main</span> </span>&#123;</span><br><span class="line">	<span class="keyword">static</span> StringBuffer stringBuffer=<span class="keyword">new</span> StringBuffer();</span><br><span class="line">	<span class="keyword">static</span> ArrayList&lt;String&gt; arrayList= <span class="keyword">new</span> ArrayList&lt;&gt;();</span><br><span class="line">	<span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>&#123;</span><br><span class="line">		Scanner scanner = <span class="keyword">new</span> Scanner(System.in);</span><br><span class="line">		<span class="keyword">char</span>[] ca=scanner.nextLine().toCharArray();</span><br><span class="line">		<span class="keyword">char</span>[] cb = scanner.nextLine().toCharArray();</span><br><span class="line">		<span class="keyword">int</span>[][] info = <span class="keyword">new</span> <span class="keyword">int</span>[cb.length+<span class="number">1</span>][ca.length+<span class="number">1</span>];</span><br><span class="line">		<span class="keyword">for</span> (<span class="keyword">int</span> b=<span class="number">0</span>;b&lt;=cb.length;b++) &#123;</span><br><span class="line">			<span class="keyword">for</span> (<span class="keyword">int</span> a=<span class="number">0</span>;a&lt;=ca.length;a++) &#123;</span><br><span class="line">				<span class="keyword">if</span> (a==<span class="number">0</span>||b==<span class="number">0</span>) &#123;</span><br><span class="line">					info[b][a]=<span class="number">0</span>;</span><br><span class="line">				&#125;<span class="keyword">else</span> <span class="keyword">if</span> (cb[b-<span class="number">1</span>]==ca[a-<span class="number">1</span>]) &#123;</span><br><span class="line">					info[b][a]=info[b-<span class="number">1</span>][a-<span class="number">1</span>]+<span class="number">1</span>;</span><br><span class="line">				&#125;<span class="keyword">else</span> &#123;</span><br><span class="line">					info[b][a]=Math.max(info[b-<span class="number">1</span>][a], info[b][a-<span class="number">1</span>]);</span><br><span class="line">				&#125;</span><br><span class="line">			&#125;</span><br><span class="line">		&#125;</span><br><span class="line">		showLongString(info, ca, cb, ca.length, cb.length);</span><br><span class="line">		<span class="comment">//输出二维矩阵   矩阵的最后一位就是最长公共子序列</span></span><br><span class="line">		<span class="keyword">for</span> (<span class="keyword">int</span> a=<span class="number">0</span>;a&lt;info.length;a++) &#123;</span><br><span class="line">			<span class="keyword">for</span> (<span class="keyword">int</span> b=<span class="number">0</span>;b&lt;info[<span class="number">0</span>].length;b++) &#123;</span><br><span class="line">				System.out.print(info[a][b]);</span><br><span class="line">			&#125;</span><br><span class="line">			System.out.println();</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">for</span> (<span class="keyword">int</span> a=<span class="number">0</span>;a&lt;arrayList.size();a++) &#123;</span><br><span class="line">			<span class="comment">//打印出所有的序列</span></span><br><span class="line">			System.out.println(arrayList.get(<span class="number">0</span>));</span><br><span class="line">		&#125;</span><br><span class="line">		scanner.close();</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="comment">//使用递归将所有的路径都找到</span></span><br><span class="line">	<span class="function"><span class="keyword">static</span> <span class="keyword">public</span> <span class="keyword">void</span> <span class="title">showLongString</span><span class="params">(<span class="keyword">int</span>[][] info,<span class="keyword">char</span>[] ca,<span class="keyword">char</span>[] cb,<span class="keyword">int</span> a,<span class="keyword">int</span> b)</span> </span>&#123;</span><br><span class="line">		<span class="keyword">if</span> (info[b][a]==<span class="number">0</span>) &#123;</span><br><span class="line">			arrayList.add(stringBuffer.reverse().toString());</span><br><span class="line">			<span class="keyword">return</span>;</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">if</span> (info[b][a]!=info[b-<span class="number">1</span>][a]&amp;&amp;info[b][a]!=info[b][a-<span class="number">1</span>]) &#123;</span><br><span class="line">			stringBuffer.append(ca[a-<span class="number">1</span>]);</span><br><span class="line">			showLongString(info, ca, cb, a-<span class="number">1</span>, b-<span class="number">1</span>);</span><br><span class="line">			stringBuffer.deleteCharAt(stringBuffer.length()-<span class="number">1</span>);</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">if</span> (info[b][a]==info[b-<span class="number">1</span>][a])&#123;</span><br><span class="line">			showLongString(info, ca, cb, a, b-<span class="number">1</span>);</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">if</span> (info[b][a]==info[b][a-<span class="number">1</span>]) &#123;</span><br><span class="line">			showLongString(info, ca, cb, a-<span class="number">1</span>, b);</span><br><span class="line">		&#125;</span><br><span class="line">	&#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<blockquote>
<p>过程</p>
</blockquote>
<ul>
<li>矩阵构建过程</li>
</ul>
<p><img src="/blogimg/suanfa/2.png" alt=""></p>
<ul>
<li>推导字符串过程</li>
</ul>
<p><img src="/blogimg/suanfa/3.png" alt=""></p>
<p>收获和反思</p>
<ol>
<li>动态规划优化的时间复杂度并不能都到O（n）</li>
<li>状态转移方程尤为重要要记住相关的数组的相关位置表示的意义（也就是状态表达式所标示的意义）</li>
<li>状态是一个非常有用的东西，在日志系统追踪和其他地方都有应用要牢记</li>
<li>递归时的意义要记住</li>
</ol>

      
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